The golden ratio is the positive solution to this equation:
x - 1 = (1/x) which is the same as
x^2 - x = 1 which is the same as
x^2 - x - 1 = 0 which is a quadratic equation

The positive solution is 1.61803...

It was discovered long ago that rectangular objects having dimensions in this ratio are pleasing to the eye.

Research has proven that this ratio manifests throughout nature in many intersting ways. The golden ratio appears to encode something fundamental to our universe.

The golden ratio is also related to the Fibonacci number sequence. That sequence is:
1, 1, 2 , 3, 5, 8, 13, 21, 34, 55.....
Where the next number is the sum of the 2 preceeding numbers. You can verify that, as the numbers become larger, the ratio of a number to the number preceeding it approaches the golden ratio exactly.